Necessary and sufficient conditions for the boundedness of weighted Hardy operators in Holder spaces

摘要

We study one- and multi-dimensional weighted Hardy operators on functions with Holder-type behavior. As a main result, we obtain necessary and sufficient conditions on the power weight under which both the left and right hand sided Hardy operators map, roughly speaking, functions with the Holder behavior only at the singular point x=0 to functions differentiable for x0 and bounded after multiplication by a power weight. As a consequence, this implies necessary and sufficient conditions for the boundedness in Holder spaces due to the corresponding imbeddings. In the multi-dimensional case we provide, in fact, stronger Hardy inequalities via spherical means. We also separately consider the case of functions with Holder-type behavior at infinity (Holder spaces on the compactified Rn).